Darth Vader acts like he doesn't even know C-3PO or R2D2. They were with him since he was a child. He did everything he could to forget his past. Also, his plating had been replaced many times, and protocol droids all look very similar.
The base assumption of why Obi-Wan doesn't remember R2-D2 -- in light of the prequels -- is that Obi-Wan is feigning ignorance in A New Hope to downplay the importance of the droid's arrival. That explanation holds the most water for a number of reasons.
R2-D2 witnessed some of Anakin's best moments and knew him when he was an innocent boy on Tatooine; it might have been healing for Luke to hear these stories. Perhaps, like the stories about Padmé, they were simply too painful to share.
R2-D2 was present when Padmé gave birth to Leia and Luke, so he knew Luke was Anakin's son – but what he didn't know was that Anakin became Darth Vader. In addition to that, Anakin's fall was surely a painful event in R2-D2's life, one that he definitely wouldn't like to remember or even talk about.
Why Didn't Darth Vader Remember C-3PO in The Empire Strikes Back? Star Wars Explained (Legends)
Who did R2-D2 originally belong to?
In The Phantom Menace, set 32 years before A New Hope, R2-D2 is portrayed as belonging to the Naboo defense forces, one of three astromech droids deployed for repair duty onboard Queen Padmé Amidala's (Natalie Portman) starship as it attempts to get past the Trade Federation blockade.
Darth Vader acts like he doesn't even know C-3PO or R2D2. They were with him since he was a child. He did everything he could to forget his past. Also, his plating had been replaced many times, and protocol droids all look very similar.
Canon characters, such as Anakin Skywalker, whose relationship with R2-D2 is one of the best in the whole series, Rey, Poe Dameron, Hera Snydulla and Kazuda Xiono are fluent in the droid language. And after spending decades with R2 himself, Luke Skywalker is also able to understand him.
Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
Leia's Force abilities and her lack of awareness of her true identity also contributed to Vader's inability to sense their familial connection. Vader's denial of Anakin Skywalker's legacy and his fear of getting hurt again prevented him from realizing the truth about his children until later in the original trilogy.
During the Clone Wars, it was common practice to wipe the minds of astromech droids after every mission to avoid the risk of Separatists taking valued information. But Anakin remained staunchly against this idea for R2-D2, meaning the droid never had to worry about losing the memories of his past.
After decades of missions with R2-D2, Luke Skywalker was reasonably fluent in droidspeak. The most common way for binary droids to communicate with organics was by writing, conveyed via another computer system. This is how astromech droid copilots would speak with the pilot of a starfighter.
When Luke Skywalker arrived on Dagobah in 3 ABY, the King of the Dragonsnakes swallowed his astromech droid R2-D2, but then spit the droid back out due to its being indigestible.
She senses Anakin when she reaches out to the pilot — and Vader realizes that Ahsoka, his former Padawan, lives on. In Season 2, Episode 18, “Shroud of Darkness,” Ahsoka's guilt manifests as a vision of Anakin, who's angry with her for leaving him all those years ago.
Darth Vader didn't learn he had a daughter until Return of the Jedi, when he stood before Emperor Palpatine. The Emperor's power probed Luke's mind, attempting to goad him to fall to the dark side, and he discovered Leia's existence. "Sister," the Emperor taunted.
They aren't abbreviations — they're model and serial numbers. R2 is a second generation R-series astromech, serial number D2. * And C-3PO is a C-series protocol droid with the serial number 3PO.
After the Clone Wars, however, Brooks Propulsion Devices, the company responsible for the development of his boosters, was shut down, and his rocket boosters were eventually damaged and thus could not be repaired. Despite this, however, R2 hardly regretted the loss of his ability to fly.
The name derives from when Lucas was making one of his earlier films, American Graffiti . Sound editor Walter Murch states that he is responsible for the utterance which sparked the name for the droid. Murch asked for Reel 2, Dialog Track 2, in the abbreviated form "R-2-D-2".
